Huber's estimator.
Let \(\beta >0\) . For all \(x\ge 0\) , let
\(\psi _H(x)=x \,\mathbb {1}\lbrace x\le \beta \rbrace +\beta \, \mathbb {1}\lbrace x>\beta \rbrace .\) 
In dimension 1, the M-estimator constructed from this score function is called Huber's estimator [1]}.
Catoni's estimator.
Let \(\beta >0\) . For all \(x\ge 0\) , let
\(\psi _C(x)=\beta \log \left(1+\frac{x}{\beta }+\frac{1}{2}\left(\frac{x}{\beta }\right)^2\right).\) 
The associated M-estimator is one of the estimators considered by Catoni in [2]}.
We call the resulting M-estimator Catoni's estimator.
Polynomial estimator.
Let \(p\in \mathbb {N}^*\) , \(\beta >0\) . For all \(x\ge 0\) , let
\(\psi _{P}(x)=\frac{x}{1+\left(\frac{x}{\beta }\right)^{1-1/p}}.\) 
We call Polynomial estimator the M-estimator obtained using this score function.

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